Compund Angle Identities and proof

So, you can split up each fraction and then apply the formulae for tangent (which you have done). After that, you can see that all the terms will cancel out, resulting in 0.In summary, the given expression can be simplified by expanding the numerators using the addition formulae and then factoring out a common term in each fraction. Then, applying the formula for tangent, all the terms will cancel out, resulting in the final answer of 0.

Homework Statement

prove using the compound angle identies, proove the following:
$$\frac{sin(A-B)}{cos(A)cos(B)}+\frac{sin(B-C)}{cos(B)cos(C)}+\frac{sin(C-A)}{cos(C)cos(A)}=0$$

n/a

The Attempt at a Solution

I resolved it to
$$\frac{sin(A)cos(B)-cos(A)sin(B)}{cos(A)cos(B)}+\frac{sin(B)cos(C)-cos(B)sin(C)}{cos(B)cos(C)}+\frac{sin(C)cos(A)-cos(C)sin(A)}{cos(C)cos(A)}=0$$

And using wolfram alpha I found that the first term resolves to tan(A)-tan(B) and then I can see how it equals zero as all the tans would cancel out.

But I have no idea how each term simplifies to that.

Any help is really appreciated.

thanks.

After you have expanded the numerators with the addition formulae (which you have done), you can simplify each fraction by taking out a factor.

In each fraction, you have something of the form

$$\frac{a+b}{c}$$

You should always check if splitting up the fraction into

$$\frac{a}{c}+\frac{b}{c}$$

1. What are Compound Angle Identities?

Compound Angle Identities are trigonometric formulas that express the values of trigonometric functions of the sum or difference of two angles in terms of the values of the trigonometric functions of those angles.

2. How do I use Compound Angle Identities in proofs?

Compound Angle Identities can be used to simplify complex trigonometric expressions and equations, making them easier to manipulate and solve in proofs. By substituting the values of the trigonometric functions of the sum or difference of two angles, we can prove the equality of two expressions.

3. What are some common Compound Angle Identities?

Some common Compound Angle Identities include the double angle identities (such as sin 2x = 2sin x cos x), the half angle identities (such as sin^2 x = (1-cos 2x)/2), and the product-to-sum identities (such as sin x cos y = (1/2)(sin(x+y) + sin(x-y))).

4. How do I derive Compound Angle Identities?

Compound Angle Identities can be derived using basic trigonometric identities (such as the Pythagorean identity and the sum and difference identities) and algebraic manipulation. It is important to understand the fundamental principles of trigonometry and have a strong foundation in algebra before attempting to derive Compound Angle Identities.

5. Why are Compound Angle Identities important?

Compound Angle Identities are important because they allow us to simplify complex trigonometric expressions and equations, making them easier to work with in proofs and real-world applications. They also help us to better understand the relationships between different trigonometric functions and angles.

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